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Theorem issubrg2 20242
Description: Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
Hypotheses
Ref Expression
issubrg2.b 𝐵 = (Base‘𝑅)
issubrg2.o 1 = (1r𝑅)
issubrg2.t · = (.r𝑅)
Assertion
Ref Expression
issubrg2 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦   𝑥, · ,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   1 (𝑥,𝑦)

Proof of Theorem issubrg2
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgsubg 20228 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2 issubrg2.o . . . 4 1 = (1r𝑅)
32subrg1cl 20230 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)
4 issubrg2.t . . . . . 6 · = (.r𝑅)
54subrgmcl 20234 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝐴𝑦𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
653expb 1120 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
76ralrimivva 3197 . . 3 (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
81, 3, 73jca 1128 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴))
9 simpl 483 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Ring)
10 simpr1 1194 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
11 eqid 2736 . . . . . . 7 (𝑅s 𝐴) = (𝑅s 𝐴)
1211subgbas 18932 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅s 𝐴)))
1310, 12syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅s 𝐴)))
14 eqid 2736 . . . . . . 7 (+g𝑅) = (+g𝑅)
1511, 14ressplusg 17171 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
1610, 15syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
1711, 4ressmulr 17188 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → · = (.r‘(𝑅s 𝐴)))
1810, 17syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (.r‘(𝑅s 𝐴)))
1911subggrp 18931 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → (𝑅s 𝐴) ∈ Grp)
2010, 19syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Grp)
21 simpr3 1196 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
22 oveq1 7364 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
2322eleq1d 2822 . . . . . . . 8 (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
24 oveq2 7365 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
2524eleq1d 2822 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
2623, 25rspc2v 3590 . . . . . . 7 ((𝑢𝐴𝑣𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
2721, 26syl5com 31 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
28273impib 1116 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
29 issubrg2.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
3029subgss 18929 . . . . . . . . . 10 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴𝐵)
3110, 30syl 17 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴𝐵)
3231sseld 3943 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢𝐴𝑢𝐵))
3331sseld 3943 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣𝐴𝑣𝐵))
3431sseld 3943 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤𝐴𝑤𝐵))
3532, 33, 343anim123d 1443 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴𝑤𝐴) → (𝑢𝐵𝑣𝐵𝑤𝐵)))
3635imp 407 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢𝐵𝑣𝐵𝑤𝐵))
3729, 4ringass 19984 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3837adantlr 713 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3936, 38syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
4029, 14, 4ringdi 19987 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4140adantlr 713 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4236, 41syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4329, 14, 4ringdir 19988 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4443adantlr 713 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4536, 44syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
46 simpr2 1195 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 1𝐴)
4732imp 407 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → 𝑢𝐵)
4829, 4, 2ringlidm 19992 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑢𝐵) → ( 1 · 𝑢) = 𝑢)
4948adantlr 713 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐵) → ( 1 · 𝑢) = 𝑢)
5047, 49syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → ( 1 · 𝑢) = 𝑢)
5129, 4, 2ringridm 19993 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑢 · 1 ) = 𝑢)
5251adantlr 713 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐵) → (𝑢 · 1 ) = 𝑢)
5347, 52syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → (𝑢 · 1 ) = 𝑢)
5413, 16, 18, 20, 28, 39, 42, 45, 46, 50, 53isringd 20009 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Ring)
5531, 46jca 512 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝐴𝐵1𝐴))
5629, 2issubrg 20222 . . . 4 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))
579, 54, 55, 56syl21anbrc 1344 . . 3 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
5857ex 413 . 2 (𝑅 ∈ Ring → ((𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRing‘𝑅)))
598, 58impbid2 225 1 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wss 3910  cfv 6496  (class class class)co 7357  Basecbs 17083  s cress 17112  +gcplusg 17133  .rcmulr 17134  Grpcgrp 18748  SubGrpcsubg 18922  1rcur 19913  Ringcrg 19964  SubRingcsubrg 20218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-subg 18925  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220
This theorem is referenced by:  opprsubrg  20243  subrgint  20244  issubrg3  20250  issubrngd2  20658  cnsubrglem  20847  mplsubrg  21411  mplind  21478  dmatsrng  21850  scmatsrng  21869  scmatsrng1  21872  cpmatsrgpmat  22070
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